Analytic aspheric coefficients to reduce the spherical aberration of lens elements used in collimated light

We provide closed-form formulas for aspheric terms for either plano-convex or convex-plano aspheric lenses as functions of the paraxial parameters involved in the process of refraction. These formulas are obtained through an expansion in Taylor's series from the exact caustic equation produced by aspheric lenses considering a plane wavefront propagating parallel to the optical axis and impinging on the refracting surface. A comparison of the aspheric coefficients obtained through our analytic formulas and commercial optical design software is presented, showing good agreement. This is useful in reducing spherical aberration.